Triple Integrals in Cylindrical Coordinates

Definition

Triple integrals in cylindrical coordinates are evaluated by mapping Cartesian space $(x, y, z)$ to $(r, \theta, z)$, where $(r, \theta)$ are polar coordinates in the $xy$-plane. $\iiint_D f(x, y, z) dV = \int_\alpha^\beta \int_{g_1(\theta)}^{g_2(\theta)} \int_{h_1(r, \theta)}^{h_2(r, \theta)} f(r \cos \theta, r \sin \theta, z) r dz dr d\theta$

• How to read: “Triple integral over D of f dV equals the iterated integral from alpha to beta, from g-one of theta to g-two of theta, from h-one of r-theta to h-two of r-theta, of f at (r cos theta, r sin theta, z) times r dz dr d-theta.”
• Meaning: Converts a volume integral in Cartesian coordinates into cylindrical form; the extra factor $r$ accounts for the polar area element in the $xy$-plane.

Why It Matters

Cylindrical coordinates are the natural choice for problems with rotational symmetry (like pipes, silos, or electromagnetic coils). Using them simplifies the limits of integration, turning a complex Cartesian nightmare into a simple, manageable calculation.

Core Concepts

• Coordinate Transformation: $x = r \cos \theta$, $y = r \sin \theta$, and $z = z$.

• How to read: “x equals r cosine theta, y equals r sine theta, z equals z.”
• Meaning: Standard cylindrical map; replace every $x,y$ in the integrand and bounds with these before integrating.

• Volume Differential: $dV = r dz dr d\theta$. The $r$ factor is inherited from the polar area differential $dA = r dr d\theta$.

• How to read: “dV equals r dz dr d-theta.”
• Meaning: The Jacobian of the coordinate change; forgetting $r$ undercounts volume away from the $z$-axis.

• Symmetry Alignment: Used when the solid $D$ is bounded by cylinders, vertical planes through the origin, or surfaces whose equations are simpler in polar form.

Connected Concepts

• Triple Integrals in Rectangular Coordinates
• Triple Integrals in Spherical Coordinates
• Moments of Inertia